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We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with diffusion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.Furthermore, we assume that F : R n × R → R is a continuous function, nonincreasing in the second variable, i.e.,This makes the operator in (1.1) degenerate parabolic.The symbol ∂W denotes the subdifferential of W . In general, the subdifferential of a convex lower semi-continuous function ϕ on a Hilbert space H endowed with a